What type of decimal expansion will (\frac{2^4\cdot 13}{2^7\cdot 5^3\cdot 13^2}) have?

Non-terminating recurring

A rational number has a terminating decimal only when, after reducing the fraction to lowest terms, its denominator has no prime factors other than 2 and 5. If any other prime remains in the denominator, the decimal division cannot end; because the remainders eventually repeat, the decimal is non-terminating recurring.

Here, cancel the common factors in the numerator and denominator: \(2^4\) cancels part of \(2^7\), and one factor 13 cancels part of \(13^2\). The reduced denominator is \(2^3\cdot 5^3\cdot 13\). Since the prime factor 13 remains, the decimal expansion is non-terminating recurring. Therefore, option B is correct; it is not terminating and cannot be non-recurring because the number is rational.

This is a Class 10 Mathematics question. Chapter: Real Numbers. Topic: Decimal expansion of rational numbers.